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doi: 10.3934/jimo.2022060
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Inertial iterative method for solving variational inequality problems of pseudo-monotone operators and fixed point problems of nonexpansive mappings in Hilbert spaces

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China

2. 

Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China

3. 

Department of Mathematics, Luoyang Normal University, Luoyang, 471022, China

*Corresponding author: Yuanheng Wang

Received  July 2021 Revised  March 2022 Early access April 2022

Fund Project: This work was supported by the NSF of China (12171435, 11971216)

In this paper, we propose a new inertial viscosity iterative algorithm for solving the variational inequality problem with a pseudo-monotone operator and the fixed point problem involving a nonexpansive mapping in real Hilbert spaces. The advantage of the proposed algorithm is that it can work without the prior knowledge of the Lipschitz constant of the mapping. The strong convergence of the sequence generated by the proposed algorithm is proved under some suitable assumptions imposed on the parameters. Some numerical experiments are given to support our main results.

Citation: Shaotao Hu, Yuanheng Wang, Bing Tan, Fenghui Wang. Inertial iterative method for solving variational inequality problems of pseudo-monotone operators and fixed point problems of nonexpansive mappings in Hilbert spaces. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022060
References:
[1]

F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space, SIAM J. Optim., 14 (2003), 773-782.  doi: 10.1137/S1052623403427859.

[2]

F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set Valued Anal., 9 (2001), 3-11.  doi: 10.1023/A:1011253113155.

[3]

R. I. BoţE. R. Csetnek and A. Heinrich, A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators, SIAM J. Optim., 23 (2013), 2011-2036.  doi: 10.1137/12088255X.

[4]

R. I. BoţE. R. CsetnekA. Heinrich and C. Hendrich, On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems, Math. Program., 150 (2015), 251-279.  doi: 10.1007/s10107-014-0766-0.

[5]

R. I. BoţE. R. Csetnek and C. Hendrich, Inertial Douglas-Rachford splitting for monotone inclusion, Appl. Math. Comput., 256 (2015), 472-487.  doi: 10.1016/j.amc.2015.01.017.

[6]

G. CaiQ. L. Dong and Y. Peng, Strong convergence theorems for solving variational inequality problems with pseudo-monotone and non-Lipschitz Operators, J. Optim. Theory Appl., 188 (2021), 447-472.  doi: 10.1007/s10957-020-01792-w.

[7]

Y. CensorA. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148 (2011), 318-335.  doi: 10.1007/s10957-010-9757-3.

[8]

Y. CensorA. Gibali and S. Reich, Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2012), 1119-1132.  doi: 10.1080/02331934.2010.539689.

[9]

Y. CensorA. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algor., 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.

[10]

P. CholamjiakD. V. Thong and Y. J. Cho, A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems, Acta Appl. Math., 169 (2020), 217-245.  doi: 10.1007/s10440-019-00297-7.

[11]

R. W. Cottle and J. C. Yao, Pseudo-monotone complementarity problems in Hilbert space, J. Optim. Theory Appl., 75 (1992), 281-295.  doi: 10.1007/BF00941468.

[12]

J. E. Denis and J. J. Morè, A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comput., 28 (1974), 549-560.  doi: 10.1090/S0025-5718-1974-0343581-1.

[13]

Q. L. DongY. J. ChoL. L. Zhong and T. M. Rassias, Inertial projection and contraction algorithms for variational inequalities, J. Glob. Optim., 70 (2018), 687-704.  doi: 10.1007/s10898-017-0506-0.

[14]

Q. L. DongK. R. KazmiK. R. AliR. Ali and X. H. Li, Inertial Krasnosel'skii-Mann type hybrid algorithms for solving hierarchical fixed point problems, J. Fixed Point Theory Appl., 21 (2019), 57. 

[15]

Q. L. DongH. B. YuanY. J. Cho and T. M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings, Optim. Lett., 12 (2018), 87-102.  doi: 10.1007/s11590-016-1102-9.

[16]

F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research. Springer-Verlag, New York, 2003.

[17]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984.

[18]

A. Gibali and D. V. Hieu, A new inertial double-projection method for solving variational inequalities, J. Fixed Point Theory Appl., 21 (2019), Paper No. 97, 21 pp. doi: 10.1007/s11784-019-0726-7.

[19]

A. GibaliD. V. Thong and P. A. Tuan, Two simple projection-type methods for solving variational inequalities, Anal. Math. Phys., 9 (2019), 2203-2225.  doi: 10.1007/s13324-019-00330-w.

[20]

D. V. HieuP. K. Anh and L. D. Muu, Modified hybrid projection methods for finding common solutions to variational inequality problems, Comput. Optim. Appl., 66 (2017), 75-96.  doi: 10.1007/s10589-016-9857-6.

[21]

X. Hu and J. Wang, Solving pseudo-monotone variational inequalities and pseudo-convex optimization problems using the projection neural network, IEEE Trans. Neural Netw., 17 (2006), 1487-1499. 

[22]

A. N. Iusem and R. Garciga Otero, Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces, Numer. Funct. Anal. Optim., 22 (2001), 609-640.  doi: 10.1081/NFA-100105310.

[23] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. 
[24]

I. V. Konnov, Combined relaxation methods for variational inequalities, Russian Math.(Iz. VUZ), 45 (2001), 43-51. 

[25]

G. M. Korpelevič, The extragradient method for finding saddle points and other problems, Ékonom. i Mat. Metody, 12 (1976), 747-756. 

[26]

R. Kraikaew and S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163 (2014), 399-412.  doi: 10.1007/s10957-013-0494-2.

[27]

P. E. Maingé, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499-1515.  doi: 10.1137/060675319.

[28]

P. E. Maingé, Convergence theorem for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236.  doi: 10.1016/j.cam.2007.07.021.

[29]

P. E. Maingé, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499-1515.  doi: 10.1137/060675319.

[30]

Y. V. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim., 25 (2015), 502-520.  doi: 10.1137/14097238X.

[31]

N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201.  doi: 10.1007/s10957-005-7564-z.

[32]

Z. Opial, Weak convergence of the successive approximations for nonexpansive mappings in Banach spaces, Bull. Amer. Math. Soc., 73 (1967), 591-597.  doi: 10.1090/S0002-9904-1967-11761-0.

[33]

S. ReichD. V. ThongQ. L. DongX. H. Li and V. T. Dung, New algorithms and convergence theorems for solving variational inequalities with non-Lipschitz mappings, Numer. Algor., 87 (2021), 527-549.  doi: 10.1007/s11075-020-00977-8.

[34]

Y. ShehuO. S. Iyiola and C. D. Enyi, An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces, Numer. Algor., 72 (2016), 835-864.  doi: 10.1007/s11075-015-0069-4.

[35]

M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765-776.  doi: 10.1137/S0363012997317475.

[36]

M. V. Solodov and P. Tseng, Modified projection-type methods for monotone variational inequalities, SIAM J. Control Optim., 34 (1996), 1814-1830.  doi: 10.1137/S0363012994268655.

[37]

B. Tan and S. Li, Strong convergence of inertial Mann algorithms for solving hierarchical fixed point problems, J. Nonlinear Var. Anal., 4 (2020), 337-355. 

[38]

B. TanL. Liu and X. Qin, Self adaptive inertial extragradient algorithms for solving bilevel pseudomonotone variational inequality problems, Jpn. J. Ind. Appl. Math., 38 (2021), 519-543.  doi: 10.1007/s13160-020-00450-y.

[39]

B. Tan, X. Qin and J. C. Yao, Strong convergence of self-adaptive inertial algorithms for solving split variational inclusion problems with applications, J. Sci. Comput., 87 (2021), Paper No. 20, 34 pp. doi: 10.1007/s10915-021-01428-9.

[40]

B. TanX. Qin and J. C. Yao, Two modified inertial projection algorithms for bilevel pseudomonotone variational inequalities with applications to optimal control problems, Numer. Algor., 88 (2021), 1757-1786.  doi: 10.1007/s11075-021-01093-x.

[41]

D. V. Thong and D. V. Hieu, Inertial extragradient algorithms for strongly pseudomonotone variational inequalities, J. Comput. Appl. Math., 341 (2018), 80-98.  doi: 10.1016/j.cam.2018.03.019.

[42]

D. V. Thong and D. V. Hieu, Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems, Numer Algor., 80 (2019), 1283-1307.  doi: 10.1007/s11075-018-0527-x.

[43]

D. V. ThongD. V. Hieu and T. M. Rassias, Self adaptive inertial subgradient extragradient algorithms for solving pseudomonotone variational inequality problems, Optim Lett., 14 (2020), 115-144.  doi: 10.1007/s11590-019-01511-z.

[44]

D. V. ThongY. Shehu and O. S. Iyiola, Weak and strong convergence theorems for solving pseudo-monotone variational inequalities with non-Lipschitz mappings, Numer. Algor., 84 (2020), 795-823.  doi: 10.1007/s11075-019-00780-0.

[45]

D. V. Thong, Y. Shehu and O. S. Iyiola, A new iterative method for solving pseudomonotone variational inequalities with non-Lipschitz operators, Comput. Appl. Math., 39 (2020), Paper No. 108, 24 pp. doi: 10.1007/s40314-020-1136-6.

[46]

P. L. Toint, Global convergence of a class of trust region methods for nonconvex minimization in Hilbert space, IMA J. Numer. Anal., 8 (1988), 231-252.  doi: 10.1093/imanum/8.2.231.

[47]

P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446.  doi: 10.1137/S0363012998338806.

[48]

P. T. Vuong and Y. Shehu, Convergence of an extragradient-type method for variational inequality with applications to optimal control problems, Numer. Algor., 81 (2019), 269-291.  doi: 10.1007/s11075-018-0547-6.

[49]

H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.

show all references

References:
[1]

F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space, SIAM J. Optim., 14 (2003), 773-782.  doi: 10.1137/S1052623403427859.

[2]

F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set Valued Anal., 9 (2001), 3-11.  doi: 10.1023/A:1011253113155.

[3]

R. I. BoţE. R. Csetnek and A. Heinrich, A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators, SIAM J. Optim., 23 (2013), 2011-2036.  doi: 10.1137/12088255X.

[4]

R. I. BoţE. R. CsetnekA. Heinrich and C. Hendrich, On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems, Math. Program., 150 (2015), 251-279.  doi: 10.1007/s10107-014-0766-0.

[5]

R. I. BoţE. R. Csetnek and C. Hendrich, Inertial Douglas-Rachford splitting for monotone inclusion, Appl. Math. Comput., 256 (2015), 472-487.  doi: 10.1016/j.amc.2015.01.017.

[6]

G. CaiQ. L. Dong and Y. Peng, Strong convergence theorems for solving variational inequality problems with pseudo-monotone and non-Lipschitz Operators, J. Optim. Theory Appl., 188 (2021), 447-472.  doi: 10.1007/s10957-020-01792-w.

[7]

Y. CensorA. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148 (2011), 318-335.  doi: 10.1007/s10957-010-9757-3.

[8]

Y. CensorA. Gibali and S. Reich, Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2012), 1119-1132.  doi: 10.1080/02331934.2010.539689.

[9]

Y. CensorA. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algor., 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.

[10]

P. CholamjiakD. V. Thong and Y. J. Cho, A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems, Acta Appl. Math., 169 (2020), 217-245.  doi: 10.1007/s10440-019-00297-7.

[11]

R. W. Cottle and J. C. Yao, Pseudo-monotone complementarity problems in Hilbert space, J. Optim. Theory Appl., 75 (1992), 281-295.  doi: 10.1007/BF00941468.

[12]

J. E. Denis and J. J. Morè, A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comput., 28 (1974), 549-560.  doi: 10.1090/S0025-5718-1974-0343581-1.

[13]

Q. L. DongY. J. ChoL. L. Zhong and T. M. Rassias, Inertial projection and contraction algorithms for variational inequalities, J. Glob. Optim., 70 (2018), 687-704.  doi: 10.1007/s10898-017-0506-0.

[14]

Q. L. DongK. R. KazmiK. R. AliR. Ali and X. H. Li, Inertial Krasnosel'skii-Mann type hybrid algorithms for solving hierarchical fixed point problems, J. Fixed Point Theory Appl., 21 (2019), 57. 

[15]

Q. L. DongH. B. YuanY. J. Cho and T. M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings, Optim. Lett., 12 (2018), 87-102.  doi: 10.1007/s11590-016-1102-9.

[16]

F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research. Springer-Verlag, New York, 2003.

[17]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984.

[18]

A. Gibali and D. V. Hieu, A new inertial double-projection method for solving variational inequalities, J. Fixed Point Theory Appl., 21 (2019), Paper No. 97, 21 pp. doi: 10.1007/s11784-019-0726-7.

[19]

A. GibaliD. V. Thong and P. A. Tuan, Two simple projection-type methods for solving variational inequalities, Anal. Math. Phys., 9 (2019), 2203-2225.  doi: 10.1007/s13324-019-00330-w.

[20]

D. V. HieuP. K. Anh and L. D. Muu, Modified hybrid projection methods for finding common solutions to variational inequality problems, Comput. Optim. Appl., 66 (2017), 75-96.  doi: 10.1007/s10589-016-9857-6.

[21]

X. Hu and J. Wang, Solving pseudo-monotone variational inequalities and pseudo-convex optimization problems using the projection neural network, IEEE Trans. Neural Netw., 17 (2006), 1487-1499. 

[22]

A. N. Iusem and R. Garciga Otero, Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces, Numer. Funct. Anal. Optim., 22 (2001), 609-640.  doi: 10.1081/NFA-100105310.

[23] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. 
[24]

I. V. Konnov, Combined relaxation methods for variational inequalities, Russian Math.(Iz. VUZ), 45 (2001), 43-51. 

[25]

G. M. Korpelevič, The extragradient method for finding saddle points and other problems, Ékonom. i Mat. Metody, 12 (1976), 747-756. 

[26]

R. Kraikaew and S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163 (2014), 399-412.  doi: 10.1007/s10957-013-0494-2.

[27]

P. E. Maingé, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499-1515.  doi: 10.1137/060675319.

[28]

P. E. Maingé, Convergence theorem for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236.  doi: 10.1016/j.cam.2007.07.021.

[29]

P. E. Maingé, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499-1515.  doi: 10.1137/060675319.

[30]

Y. V. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim., 25 (2015), 502-520.  doi: 10.1137/14097238X.

[31]

N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201.  doi: 10.1007/s10957-005-7564-z.

[32]

Z. Opial, Weak convergence of the successive approximations for nonexpansive mappings in Banach spaces, Bull. Amer. Math. Soc., 73 (1967), 591-597.  doi: 10.1090/S0002-9904-1967-11761-0.

[33]

S. ReichD. V. ThongQ. L. DongX. H. Li and V. T. Dung, New algorithms and convergence theorems for solving variational inequalities with non-Lipschitz mappings, Numer. Algor., 87 (2021), 527-549.  doi: 10.1007/s11075-020-00977-8.

[34]

Y. ShehuO. S. Iyiola and C. D. Enyi, An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces, Numer. Algor., 72 (2016), 835-864.  doi: 10.1007/s11075-015-0069-4.

[35]

M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765-776.  doi: 10.1137/S0363012997317475.

[36]

M. V. Solodov and P. Tseng, Modified projection-type methods for monotone variational inequalities, SIAM J. Control Optim., 34 (1996), 1814-1830.  doi: 10.1137/S0363012994268655.

[37]

B. Tan and S. Li, Strong convergence of inertial Mann algorithms for solving hierarchical fixed point problems, J. Nonlinear Var. Anal., 4 (2020), 337-355. 

[38]

B. TanL. Liu and X. Qin, Self adaptive inertial extragradient algorithms for solving bilevel pseudomonotone variational inequality problems, Jpn. J. Ind. Appl. Math., 38 (2021), 519-543.  doi: 10.1007/s13160-020-00450-y.

[39]

B. Tan, X. Qin and J. C. Yao, Strong convergence of self-adaptive inertial algorithms for solving split variational inclusion problems with applications, J. Sci. Comput., 87 (2021), Paper No. 20, 34 pp. doi: 10.1007/s10915-021-01428-9.

[40]

B. TanX. Qin and J. C. Yao, Two modified inertial projection algorithms for bilevel pseudomonotone variational inequalities with applications to optimal control problems, Numer. Algor., 88 (2021), 1757-1786.  doi: 10.1007/s11075-021-01093-x.

[41]

D. V. Thong and D. V. Hieu, Inertial extragradient algorithms for strongly pseudomonotone variational inequalities, J. Comput. Appl. Math., 341 (2018), 80-98.  doi: 10.1016/j.cam.2018.03.019.

[42]

D. V. Thong and D. V. Hieu, Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems, Numer Algor., 80 (2019), 1283-1307.  doi: 10.1007/s11075-018-0527-x.

[43]

D. V. ThongD. V. Hieu and T. M. Rassias, Self adaptive inertial subgradient extragradient algorithms for solving pseudomonotone variational inequality problems, Optim Lett., 14 (2020), 115-144.  doi: 10.1007/s11590-019-01511-z.

[44]

D. V. ThongY. Shehu and O. S. Iyiola, Weak and strong convergence theorems for solving pseudo-monotone variational inequalities with non-Lipschitz mappings, Numer. Algor., 84 (2020), 795-823.  doi: 10.1007/s11075-019-00780-0.

[45]

D. V. Thong, Y. Shehu and O. S. Iyiola, A new iterative method for solving pseudomonotone variational inequalities with non-Lipschitz operators, Comput. Appl. Math., 39 (2020), Paper No. 108, 24 pp. doi: 10.1007/s40314-020-1136-6.

[46]

P. L. Toint, Global convergence of a class of trust region methods for nonconvex minimization in Hilbert space, IMA J. Numer. Anal., 8 (1988), 231-252.  doi: 10.1093/imanum/8.2.231.

[47]

P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446.  doi: 10.1137/S0363012998338806.

[48]

P. T. Vuong and Y. Shehu, Convergence of an extragradient-type method for variational inequality with applications to optimal control problems, Numer. Algor., 81 (2019), 269-291.  doi: 10.1007/s11075-018-0547-6.

[49]

H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.

Figure 1.  The behavior of our Algorithm 3.1 in Example 4.1 ($ m = 200 $)
Figure 2.  The behavior of our Algorithm 3.1 in Example 4.2 ($ x_{0} = x_{1} = 10\exp(t) $)
Figure 3.  The behavior of our Algorithm 3.1 in Example 4.3 ($ m = 500000 $)
Table 1.  Numerical results of all algorithms with different dimensions in Example 4.1
Algorithms $ m=20 $ $ m=50 $ $ m=100 $ $ m=200 $
$ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU
Our Alg. 3.1 4.47E-05 0.0303 1.70E-04 0.0316 3.17E-04 0.0707 5.00E-04 0.1119
CTC Alg. 3.1 3.36E-04 0.0227 9.95E-04 0.0227 1.69E-03 0.0250 2.64E-03 0.0273
GTT Alg. 3.1 2.80E-03 0.0310 6.86E-03 0.0361 1.08E-02 0.1044 1.63E-02 0.1393
GTT Alg. 3.2 3.38E-03 0.0483 9.78E-03 0.0379 1.64E-02 0.0704 2.56E-02 0.1027
Algorithms $ m=20 $ $ m=50 $ $ m=100 $ $ m=200 $
$ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU
Our Alg. 3.1 4.47E-05 0.0303 1.70E-04 0.0316 3.17E-04 0.0707 5.00E-04 0.1119
CTC Alg. 3.1 3.36E-04 0.0227 9.95E-04 0.0227 1.69E-03 0.0250 2.64E-03 0.0273
GTT Alg. 3.1 2.80E-03 0.0310 6.86E-03 0.0361 1.08E-02 0.1044 1.63E-02 0.1393
GTT Alg. 3.2 3.38E-03 0.0483 9.78E-03 0.0379 1.64E-02 0.0704 2.56E-02 0.1027
Table 2.  Numerical results of all algorithms with different initial values in Example 4.2
Algorithms $ x_{1}=10t^{3} $ $ x_{1}=10\sin(2t) $ $ x_{1}=10\log(t) $ $ x_{1}=10\exp(t) $
$ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU
Our Alg. 3.1 1.47E-15 37.7392 4.04E-15 38.5818 7.26E-15 39.9932 3.16E-15 44.6711
CTC Alg. 3.1 3.16E-13 22.8652 2.55E-13 24.3823 3.36E-12 25.0381 4.39E-12 30.0801
GTT Alg. 3.1 9.04E-06 33.7345 1.54E-05 34.5784 2.31E-05 36.9545 2.01E-05 45.1488
GTT Alg. 3.2 6.25E-11 33.5898 7.40E-10 36.0191 1.04E-09 37.5059 1.06E-09 44.7099
Algorithms $ x_{1}=10t^{3} $ $ x_{1}=10\sin(2t) $ $ x_{1}=10\log(t) $ $ x_{1}=10\exp(t) $
$ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU
Our Alg. 3.1 1.47E-15 37.7392 4.04E-15 38.5818 7.26E-15 39.9932 3.16E-15 44.6711
CTC Alg. 3.1 3.16E-13 22.8652 2.55E-13 24.3823 3.36E-12 25.0381 4.39E-12 30.0801
GTT Alg. 3.1 9.04E-06 33.7345 1.54E-05 34.5784 2.31E-05 36.9545 2.01E-05 45.1488
GTT Alg. 3.2 6.25E-11 33.5898 7.40E-10 36.0191 1.04E-09 37.5059 1.06E-09 44.7099
Table 3.  Numerical results of all algorithms with different dimensions in Example 4.3
Algorithms $ m=500 $ $ m=5000 $ $ m=50000 $ $ m=500000 $
$ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU
Our Alg. 3.1 7.13E-57 0.0249 8.76E-57 0.1079 3.77E-57 0.4058 8.89E-57 13.8430
CDP Alg. 3.1 3.97E-27 0.0406 7.89E-27 0.1274 7.25E-27 0.5290 4.78E-26 13.9558
TSI Alg. 3 8.38E-13 0.0318 7.96E-13 0.1270 8.17E-13 0.4180 6.62E-13 15.0426
RTDLD Alg. 4 4.72E-10 0.0312 3.07E-07 0.1132 1.64E-03 0.4540 2.59E-02 19.2562
Algorithms $ m=500 $ $ m=5000 $ $ m=50000 $ $ m=500000 $
$ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU
Our Alg. 3.1 7.13E-57 0.0249 8.76E-57 0.1079 3.77E-57 0.4058 8.89E-57 13.8430
CDP Alg. 3.1 3.97E-27 0.0406 7.89E-27 0.1274 7.25E-27 0.5290 4.78E-26 13.9558
TSI Alg. 3 8.38E-13 0.0318 7.96E-13 0.1270 8.17E-13 0.4180 6.62E-13 15.0426
RTDLD Alg. 4 4.72E-10 0.0312 3.07E-07 0.1132 1.64E-03 0.4540 2.59E-02 19.2562
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